I have a homework problem which consists of two parts, the first of which I have been staring at for several days with very little (constructive) progress.

I need to show that the function $$f(t) = \sum_{n=1}^{\infty}\frac{\cos(3^{n}t)}{3^{n\alpha}}\in\Lambda^{\alpha}$$ when $0 < \alpha \leq 1$. The second part is to show that if $\alpha < \beta < 1$, then $f\notin\Lambda^{\beta}$, but I'll worry about the second part later.

EDIT: Removed the last half - dozen lines which turned out to be completely non-constructive.

Now I'm not sure if I'm even remotely close to going down the right path, but if I could get this manipulated into something of the form $C^{\alpha}$ I'd be done. But I just can't seem to go any further. Any suggestions?

1 Answer
1

Use the Mean Value Theorem on the terms with $n<N$:
$$
\begin{align}
|f(t+h)-f(t)|&\le\sum_{n=1}^\infty\left|\frac{\cos(3^n(t+h))-\cos(3^nt)}{3^{n\alpha}}\right|\\
&=\sum_{n=1}^{N-1}\left|\frac{3^nh\sin(3^n(t+\eta_n))}{3^{n\alpha}}\right|+\sum_{n=N}^\infty\left|\frac{\cos(3^n(t+h))-\cos(3^nt)}{3^{n\alpha}}\right|\\
&\le|h|\frac{3^{N(1-\alpha)}-3^{1-\alpha}}{3^{1-\alpha}-1}+2\frac{\frac{1}{3^{N\alpha}}}{1-\frac{1}{3^{\alpha}}}\tag{1}
\end{align}
$$
Choose $N$ so that $|h|\sim3^{-N}$. Then $|h|3^{N(1-\alpha)}\sim|h|^\alpha$ and $\frac{1}{3^{N\alpha}}\sim|h|^\alpha$. Thus, the right side of $(1)\sim|h|^\alpha$.

To be more precise, let $N=\lfloor\log_3(\frac{1}{h})\rfloor$. Then $|h|3^{N(1-\alpha)}\le|h|^\alpha$ and $\frac{1}{3^{N\alpha}}\le3|h|^\alpha$. Thus,
$$
|f(t+h)-f(t)|\le\left(\frac{1}{3^{1-\alpha}-1}+\frac{6}{1-\frac{1}{3^{\alpha}}}\right)|h|^\alpha\tag{2}
$$
Note that the $\Lambda_\alpha$-norm in $(2)$ blows up near $\alpha=0$ and $\alpha=1$.

@Kyle: are you summing from $n=0$ or $n=1$?
–
robjohn♦Oct 7 '11 at 0:13

1

@Kyle: if something is less than $\displaystyle\frac{3^{N(1-\alpha)}-3^{1-\alpha}}{3^{1-\alpha}-1}$ it is certainly less than $\displaystyle\frac{3^{N(1-\alpha)}}{3^{1-\alpha}-1}$, so there is really nothing with which to deal.
–
robjohn♦Oct 7 '11 at 1:45